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mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
field of
order theory Order theory is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geom ...
, an element ''a'' of a
partially ordered set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
with
least element In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
0 is an atom if 0 < ''a'' and there is no ''x'' such that 0 < ''x'' < ''a''. Equivalently, one may define an atom to be an element that is minimal among the non-zero elements, or alternatively an element that covers the least element 0.


Atomic orderings

Let <: denote the cover relation in a partially ordered set. A partially ordered set with a least element 0 is atomic if every element ''b'' > 0 has an atom ''a'' below it, that is, there is some ''a'' such that ''b'' ≥ ''a'' :> ''0''. Every finite partially ordered set with 0 is atomic, but the set of nonnegative
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s (ordered in the usual way) is not atomic (and in fact has no atoms). A partially ordered set is relatively atomic (or ''strongly atomic'') if for all ''a'' < ''b'' there is an element ''c'' such that ''a'' <: ''c'' ≤ ''b'' or, equivalently, if every interval 'a'', ''b''is atomic. Every relatively atomic partially ordered set with a least element is atomic. Every finite poset is relatively atomic. A partially ordered set with least element 0 is called atomistic if every element is the
least upper bound In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of a set of atoms. The linear order with three elements is not atomistic (see Fig. 2). Atoms in partially ordered sets are abstract generalizations of singletons in
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...
(see Fig. 1). Atomicity (the property of being atomic) provides an abstract generalization in the context of
order theory Order theory is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geom ...
of the ability to select an element from a non-empty set.


Coatoms

The terms ''coatom'', ''coatomic'', and ''coatomistic'' are defined dually. Thus, in a partially ordered set with
greatest element In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined duality (order theory), dually, t ...
1, one says that * a coatom is an element covered by 1, * the set is coatomic if every ''b'' < 1 has a coatom ''c'' above it, and * the set is coatomistic if every element is the
greatest lower bound are equal. Image:Supremum illustration.svg, 250px, A set ''A'' of real numbers (blue circles), a set of upper bounds of ''A'' (red diamond and circles), and the smallest such upper bound, that is, the supremum of ''A'' (red diamond). In mathematic ...
of a set of coatoms.


References

*


External links

* * {{planetmath reference, id=132, title=Poset Order theory